Optimal. Leaf size=204 \[ -\frac {\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{9/2}}-\frac {5 d \sqrt {a+b x} (11 b c-21 a d)}{12 c^4 \sqrt {c+d x}}-\frac {d \sqrt {a+b x} (23 b c-35 a d)}{12 c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x} (5 b c-7 a d)}{4 c^2 x (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {98, 151, 152, 12, 93, 208} \begin {gather*} -\frac {\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{9/2}}-\frac {5 d \sqrt {a+b x} (11 b c-21 a d)}{12 c^4 \sqrt {c+d x}}-\frac {d \sqrt {a+b x} (23 b c-35 a d)}{12 c^3 (c+d x)^{3/2}}-\frac {\sqrt {a+b x} (5 b c-7 a d)}{4 c^2 x (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 151
Rule 152
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{x^3 (c+d x)^{5/2}} \, dx &=-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {\int \frac {-\frac {1}{2} a (5 b c-7 a d)-b (2 b c-3 a d) x}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 c}\\ &=-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}+\frac {\int \frac {\frac {1}{4} a \left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right )-a b d (5 b c-7 a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{2 a c^2}\\ &=-\frac {d (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {\int \frac {-\frac {3}{8} a (b c-a d) \left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right )+\frac {1}{4} a b d (23 b c-35 a d) (b c-a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 a c^3 (b c-a d)}\\ &=-\frac {d (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {5 d (11 b c-21 a d) \sqrt {a+b x}}{12 c^4 \sqrt {c+d x}}+\frac {2 \int \frac {3 a (b c-a d)^2 \left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right )}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a c^4 (b c-a d)^2}\\ &=-\frac {d (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {5 d (11 b c-21 a d) \sqrt {a+b x}}{12 c^4 \sqrt {c+d x}}+\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 c^4}\\ &=-\frac {d (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {5 d (11 b c-21 a d) \sqrt {a+b x}}{12 c^4 \sqrt {c+d x}}+\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 c^4}\\ &=-\frac {d (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {5 d (11 b c-21 a d) \sqrt {a+b x}}{12 c^4 \sqrt {c+d x}}-\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 171, normalized size = 0.84 \begin {gather*} -\frac {-x^2 \left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \left (\sqrt {c} \sqrt {a+b x} (4 a c+3 a d x+b c x)-3 a^{3/2} (c+d x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )+3 c^{5/2} x (a+b x)^{5/2} (b c-7 a d)+6 a c^{7/2} (a+b x)^{5/2}}{12 a^2 c^{9/2} x^2 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 295, normalized size = 1.45 \begin {gather*} \frac {\left (-35 a^2 d^2+30 a b c d-3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 \sqrt {a} c^{9/2}}+\frac {(a+b x)^{3/2} \left (\frac {105 a^3 d^2 (c+d x)^3}{(a+b x)^3}-\frac {175 a^2 c d^2 (c+d x)^2}{(a+b x)^2}-\frac {90 a^2 b c d (c+d x)^3}{(a+b x)^3}-\frac {15 b^2 c^3 (c+d x)^2}{(a+b x)^2}+\frac {9 a b^2 c^2 (c+d x)^3}{(a+b x)^3}-\frac {48 b c^3 d (c+d x)}{a+b x}+\frac {56 a c^2 d^2 (c+d x)}{a+b x}+\frac {150 a b c^2 d (c+d x)^2}{(a+b x)^2}+8 c^3 d^2\right )}{12 c^4 (c+d x)^{3/2} \left (c-\frac {a (c+d x)}{a+b x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 10.42, size = 634, normalized size = 3.11 \begin {gather*} \left [\frac {3 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{3} + {\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (6 \, a^{2} c^{4} + 5 \, {\left (11 \, a b c^{2} d^{2} - 21 \, a^{2} c d^{3}\right )} x^{3} + 2 \, {\left (39 \, a b c^{3} d - 70 \, a^{2} c^{2} d^{2}\right )} x^{2} + 3 \, {\left (5 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a c^{5} d^{2} x^{4} + 2 \, a c^{6} d x^{3} + a c^{7} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{3} + {\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (6 \, a^{2} c^{4} + 5 \, {\left (11 \, a b c^{2} d^{2} - 21 \, a^{2} c d^{3}\right )} x^{3} + 2 \, {\left (39 \, a b c^{3} d - 70 \, a^{2} c^{2} d^{2}\right )} x^{2} + 3 \, {\left (5 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (a c^{5} d^{2} x^{4} + 2 \, a c^{6} d x^{3} + a c^{7} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 11.76, size = 1234, normalized size = 6.05
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 679, normalized size = 3.33 \begin {gather*} -\frac {\sqrt {b x +a}\, \left (105 a^{2} d^{4} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-90 a b c \,d^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+9 b^{2} c^{2} d^{2} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+210 a^{2} c \,d^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-180 a b \,c^{2} d^{2} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+18 b^{2} c^{3} d \,x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+105 a^{2} c^{2} d^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-90 a b \,c^{3} d \,x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+9 b^{2} c^{4} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,d^{3} x^{3}+110 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c \,d^{2} x^{3}-280 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \,d^{2} x^{2}+156 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{2} d \,x^{2}-42 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,c^{2} d x +30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{3} x +12 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,c^{3}\right )}{24 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}} c^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}}{x^3\,{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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